"A luminous star, of the same density as the Earth, and whose diameter should be two hundred and fifty times larger than that of the Sun, would not, in consequence of its attraction, allow any of its rays to arrive at us; it is therefore possible that the largest luminous bodies in the universe may, through this cause, be invisible." -- Pierre Laplace, The System of the World, Book 5, Chapter VI (1798).

Evolution of Stars

1. Clouds of Hydrogen begin condensing into more dense clusters due to gravitation. 2. Eventually the density gets high enough that the Hydrogen begins fusing into Helium. This fusion releases energy, mostly in the form of electromagnetic radiation. Our sun is currently in this phase. Note that the gravitational attraction of the matter of the star is trying to make it smaller; this is balanced by the radiation pressure that is trying to push the matter outward making the star bigger. In class we showed a photograph of a birthplace of stars; the URL is http://www.seds.org/hst/M16WF2.html. After about a billion years the Hydrogen fuel is exhausted. It starts fusing the Helium into heavier elements. The temperature of the star goes up, causing it to become much bigger. In this phase the star is called a "red giant."

When this "fuel" is exhausted, there are three possible outcomes, depending on the total mass of the star:

1. If the mass of the star is less than about 1.2 solar masses, the star becomes a spinning ball of dead slag, gradually cooling down as it radiates away its thermal energy. It is called a "white dwarf." Its radius is on the order of 5000 km, and the density is about 1 ton per cubic centimeter. 2. If the mass of the star is greater than about 1.2 solar masses but less than about 3 solar masses, it goes through the white dwarf phase, but the gravitational attraction is so strong that the protons and electrons get fused into neutrons. Thus we end up with a spinning ball of neutrons, a "neutron star." The radius is on the order of 10 km, and the density is about 100 million tons per cubic centimeter. Astronomers believe that they have discovered many of these neutron stars; one of the best known is in the Crab nebula and is the remnant of a supernova that was seen on Earth in year 1054 of the current era. They "see" them because as they spin they emit a sweep of radiation, which we observe has having a regular periodicity. Thus before we knew they were neutron stars they were called "pulsars."

Another Approach to Black Holes

Above I characterized a black hole as an object whose gravitational pressure is so intense that the matter of the object is crushed out of existence. Here is another way of characterizing the same phenomenon.

Above is the embedding diagram of a normal star. Of course, the more massive the star , the deeper the "well" in the centre.

Here is the embedding diagram of a black hole. The gravitation is so intense that it has punched a hole in the fabric of space time. We call the hole in space-time a singularity.

The Wormhole Solution

Einstein did not like the singularity in the centre of the black hole. In 1935 he and Rosen found another solution to the equations of a black hole. It is shown to the above. There is an Einstein-Rosen bridge or wormhole connecting two different regions of space time.

The Einstein-Rosen solution does not say anything about the relationship between the two regions of space time. One possibility is shown above. The black hole in some region of space-time connects to a black hole in another region of space-time. The previous figure can be a bit mis-leading in trying to interpret distances. Here is a topologically equivalent version of the first one.

Finally, a sort of wild speculation. The Klein bottle shown above may be a model for how a black hole in the universe can connect to the universe as a whole.

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